The hard edge refers to the distribution of the smallest singular value for certain ensembles of random matrices, or, and what is the same, that of the minimal point of a logarithmic gas constrained to the positive half line. For any "inverse temperature" and “quadratic" potential the possible limit laws (as the dimension, or number of particles, tends to infinity) was characterized by Jose Ramirez and myself in terms of the spectrum of a (random) diffusion generator. Here we show this picture persists for more general convex polynomial potentials. Joint work with Patrick Waters.

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<b><span style="color:red"> Please note the unusual day and time </span></b>

From networks to genomics, large amounts of data are increasingly available and play critical roles in helping us understand complex systems. Statistical inference is crucial in discovering the underlying structures present in these systems, whether this concerns the time evolution of a network, an underlying geometric structure, or reconstructing a DNA sequence from partial and noisy information. In this talk I will discuss several fundamental detection and estimation problems in these areas.

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We investigate a simple model for social learning with two characters: a teacher and a student. The teacher's goal is to teach the student the state of the world <math>\Theta</math>, however, the teacher herself is not certain about <math>\Theta</math> and needs to simultaneously learn it and teach it. We examine several natural strategies the teacher may employ to make the student learn as fast as possible. Our primary technical contribution is analyzing the exact learning rates for these strategies by studying the large deviation properties of the sign of a transient random walk on <math>\mathbb Z</math>.

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I will present an overview of recent developments in source detection and estimation in randomly growing graphs. For example, can one detect the influence of the initial seed graph? How good are root-finding algorithms? I will also discuss inference in random geometric graphs: can one detect and estimate an underlying high-dimensional geometric structure? Finally, I will discuss statistical error correction algorithms for DNA sequencing that are motivated by DNA storage, which aims to use synthetic DNA as a high-density, durable, and easy-to-manipulate storage medium of digital data.

The KPZ fixed point is the Markov process at the centre of the KPZ universality class. In the talk we describe the exact solution of the totally asymmetric simple exclusion process, which is one of the models in the KPZ universality class, and provide a description of the KPZ fixed point in the 1:2:3 scaling limit. This is a joint work with Jeremy Quastel and Daniel Remenik.

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Title: '''The endpoint distribution of directed polymers'''

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Abstract: On the d-dimensional integer lattice, directed polymers are paths of a random walk in random environment, except that the environment updates at each time step. The result is a statistical mechanical system, whose qualitative behavior is governed by a temperature parameter and the law of the environment. Historically, the phase transitions have been best understood by whether or not the path’s endpoint localizes. While the endpoint is no longer a Markov process as in a random walk, its quenched distribution is. The key difficulty is that the space of measures is too large for one to expect convergence results. By adapting methods recently used by Mukherjee and Varadhan, we develop a compactification theory to resolve the issue. In this talk, we will discuss this intriguing abstraction, as well as new concrete theorems it allows us to prove for directed polymers.

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== Thursday, November 9, 2017, Chen Jia, University of Texas at Dallas ==

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This talk is based on joint work with Sourav Chatterjee.

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'''Mathematical foundation of nonequilibrium fluctuation-dissipation theorems and a biological application'''

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== Thursday, 2/2/2017, TBA ==

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The fluctuation-dissipation theorem (FDT) for equilibrium states is one of the classical results in equilibrium statistical physics. In recent years, many efforts have been devoted to generalizing the classical FDT to systems far from equilibrium. This was considered as one of the most significant progress of nonequilibrium statistical physics over the past two decades. In this talk, I will introduce our recent work on the rigorous mathematical foundation of the nonequilibrium FDTs for inhomogeneous diffusion processes and inhomogeneous continuous-time Markov chains. I will also talk about the application of the nonequilibrium FDTs to a practical biological problem called sensory adaptation.

A heat exchanger can be modeled as a closed domain containing an incompressible fluid. The fluid has some temperature distribution obeying the advection-diffusion equation, with zero temperature boundary conditions at the walls. The goal is then to start from some initial positive heat distribution, and to flux it through the walls as fast as possible. Even for a steady flow, this is a time-dependent problem, which can be hard to optimize. Instead, we consider the mean exit time of Brownian particles starting from inside the domain. A flow favorable to heat exchange should lower the exit time, and so we minimize some norm of the exit time over incompressible flows (drifts) with a given energy. This is a simpler, time-independent optimization problem, which we then proceed to solve analytically in some limits, and numerically otherwise.

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== Thursday, March 2, No Seminar this week ==

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The talk by [http://people.maths.ox.ac.uk/woolley/ Thomas Woolley], [https://www.maths.ox.ac.uk/ Oxford] has been moved to April 6 (see below).

The Sherrington-Kirkpatirck (SK) model is a mean-field spin glass introduced by theoretical physicists in order to explain the strange behavior of certain alloy, such as CuMn. Despite of its seemingly simple formulation, it was conjectured to possess a number of fruitful properties. This talk will be focused on the energy landscape of the SK model. First, we will present a formula for the maximal energy in Parisi’s formulation. Second, we will give a description of the energy landscape by showing that near any given energy level between zero and maximal energy, there exist exponentially many equidistant spin configurations. Based on joint works with Auffinger, Handschy, and Lerman.

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Interacting particle models are often employed to gain understanding of the emergence of macroscopic phenomena from microscopic laws of nature. These individual-based models capture fine details, including randomness and discreteness of individuals, that are not considered in continuum models such as partial differential equations (PDE) and integral-differential equations. The challenge, which is fundamental in any multi-scale modeling approach for complex systems, is how to simultaneously retain key information in microscopic models as well as efficiency and robustness of macroscopic models.

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== Thursday, March 23, Spring Break ==

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In this talk, I will discuss how this challenge can be overcome by elucidating the probabilistic connections between particle models and PDE, in particular, why naively adding diffusion terms to ordinary differential equations might fail to account for spatial dynamics in population models. These connections also explain how stochastic partial differential equations (SPDE) arise naturally under a suitable choice of level of detail in modeling complex systems. I will also present some novel scaling limits including SPDE on graphs and coupled SPDE. These SPDE not only interpolate between particle models and PDE, but also quantify the source and the order of magnitude of stochasticity. Scaling limit theorems and new duality formulas are obtained for these SPDE, which connect phenomena across scales and offer insights about the genealogies and the time-asymptotic properties of certain population dynamics.

Abstract: We study the problem of identifying the source of a diffusion spreading over a regular tree. When the degree of each node is at least three, we show that it is possible to construct confidence sets for the diffusion source with size independent of the number of infected nodes. Our estimators are motivated by analogous results in the literature concerning identification of the root node in preferential attachment and uniform attachment trees. At the core of our proofs is a probabilistic analysis of Polya urns corresponding to the number of uninfected neighbors in specific subtrees of the infection tree. We also describe extensions of our results to diffusions spreading over Galton-Watson trees. This is joint work with Justin Khim (UPenn).

Abstract: Mathematical models for chemical reaction networks are widely used in biochemistry, as well as in other fields. The original aim of the models is to predict the dynamics of a collection of reactants that undergo chemical transformations. There exist two standard modeling regimes: a deterministic and a stochastic one. These regimes are chosen case by case in accordance to what is believed to be more appropriate. It is natural to wonder whether the dynamics of the two different models are linked, and whether properties of one model can shed light on the behavior of the other one. Some connections between the two modelling regimes have been known for forty years, and new ones have been pointed out recently. However, many open questions remain, and the issue is still largely unexplored.

continuous initial data. The argument utilized the properties of the HJ semi-group, in particular, the finite speed of propagation. The

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last property is lost for viscous HJ equations. We remark that the above mentioned implication holds under natural conditions for both

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viscous and non-viscous Hamilton-Jacobi equations. As an application of our result, we show homogenization in a stationary ergodic setting for a special class of viscous HJ equations with a non-convex Hamiltonian in one space dimension.

Title: '''On the maximum of the characteristic polynomial of the Circular Beta Ensemble'''

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In this talk, we present our result on the extremal values of (the logarithm of) the characteristic polynomial of a random unitary matrix whose spectrum is distributed according to the Circular Beta Ensemble. Using different techniques, it gives an improvement and a generalization of the previous recent results by Arguin, Belius, Bourgade on the one hand, and Paquette, Zeitouni on the other hand. They recently treated the CUE case, which corresponds to beta equal to 2.

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== Thursday, October 6, No Seminar ==

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== Thursday, October 13, No Seminar due to [http://sites.math.northwestern.edu/mwp/ Midwest Probability Colloquium] ==

Title: Current Fluctuations of the Stationary ASEP and Six-Vertex Model

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Abstract: We consider the following three models from statistical mechanics: the asymmetric simple exclusion process, the stochastic six-vertex model, and the ferroelectric symmetric six-vertex model. It had been predicted by the physics communities for some time that the limiting behavior for these models, run under certain classes of translation-invariant (stationary) boundary data, are governed by the large-time statistics of the stationary Kardar-Parisi-Zhang (KPZ) equation. The purpose of this talk is to explain these predictions in more detail and survey some of our recent work that verifies them.

Abstract: I will describe why it is hard to do homogenization for non-convex Hamilton-Jacobi equations and explain some recent results in this direction. I will also make a very brief connection to first passage percolation and address some challenging questions which appear in both directions. This is based on joint work with Qian and Yu.

Title: '''Large deviations for irreducible Markov chains with general state space'''

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We study the large deviation principle for the empirical measure of general irreducible Markov chains in the tao topology for a broad class of initial distributions. The roles of several rate functions, including the rate function based on the convergence parameter of the transform kernel and the Donsker-Varadhan rate function, are clarified.

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Nonintersecting Brownian bridges on the unit circle form a determinantal point process whose kernel is expressed in terms of a system of discrete orthogonal polynomials which may be studied using Riemann--Hilbert techniques. If the Brownian motions have a drift, then the weight of the orthogonal polynomials becomes complex. I will discuss the tacnode and k-tacnode processes, which are related to the Painleve II function, as scaling limits of Nonintersecting Brownian motions on the unit circle and will discuss some of the features and difficulties of Riemann--Hilbert analysis of discrete orthogonal polynomials with varying complex weights.

These models, however, ignore the stochasticity and individuality of many complex systems in nature. Recognizing this drawback, scientists are developing individual-based models for model selection purposes. The latter models are sometimes studied under the framework of interacting particle systems (IPS) by mathematicians, who prove scaling limit theorems to connect various IPS with RDE across scales.

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In this talk, I will present some new limiting objects including SPDE on metric graphs and coupled SPDE. These SPDE reduce to RDE when the noise parameter tends to zero, therefore interpolates between IPS and RDE and identifies the source of stochasticity. Scaling limit theorems and novel duality formulas are obtained for these SPDE, which not only connect phenomena across scales, but also offer insights about the genealogies and time-asymptotic properties of certain population dynamics. In particular, I will present rigorous results about the lineage dynamics for of a biased voter model introduced by Hallatschek and Nelson (2007).

Title: '''On scaling limits of Open ASEP and Glauber dynamics of ferromagnetic models'''

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We discuss two scaling limit results for discrete models converging to stochastic PDEs. The first is the asymmetric simple exclusion process in contact with sources and sinks at boundaries, called Open ASEP. We prove that under weakly asymmetric scaling the height function converges to the KPZ equation with Neumann boundary conditions. The second is the Glauber dynamics of the Blume-Capel model (a generalization of Ising model), in two dimensions with Kac potential. We prove that the averaged spin field converges to the stochastic quantization equations. A common challenge in the proofs is how to identify the limiting process as the solution to the SPDE, and we will discuss how to overcome the difficulties in the two cases.(Based on joint works with Ivan Corwin and Hendrik Weber.)

Title: '''Singular Stochastic Partial Differential Equations - How do they arise and what do they mean?'''

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Abstract: Systems with random fluctuations are ubiquitous in the real world. Stochastic PDEs are default models for these random systems, just as PDEs are default models for deterministic systems. However, a large class of such stochastic PDEs were poorly understood until very recently: the presence of very singular random forcing as well as nonlinearities render it challenging to interpret what one even means by a ``solution". The recent breakthroughs by M. Hairer, M. Gubinelli and other researchers including the speaker not only established solution theories for these singular SPDEs, but also led to an explosion of new questions. These include scaling limits of random microscopic models, development of numerical schemes, ergodicity of random dynamical systems and a new approach to quantum field theory. In this talk we will discuss the main ideas of the recent solution theories of singular SPDEs, and how these SPDEs arise as limits of various important physical models.

Title: '''On some concrete criteria for quantum and stochastic confinement'''

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Abstract: In this talk we will present several recent results on criteria ensuring the confinement of a quantum or a stochastic particle to a bounded domain in <math>\mathbb{R}^n</math>. These criteria are given in terms of explicit growth and/or decay rates for the diffusion matrix and the drift potential close to the boundary of the domain. As an application of the general method, we will discuss several cases, including some where the background Riemannian manifold (induced by the diffusion matrix) is geodesically incomplete. These results are part of an ongoing joint project with G. Nenciu (IMAR, Bucharest, Romania).

'''Note:''' Daniele Cappelletti is speaking in the [http://www.math.wisc.edu/wiki/index.php/Applied/ACMS Applied Math Seminar], but his research on stochastic reaction networks uses probability theory and is related to work of our own [http://www.math.wisc.edu/~anderson/ David Anderson].

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Title: '''Deterministic and Stochastic Reaction Networks'''

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Abstract: Mathematical models of biochemical reaction networks are of great interest for the analysis of experimental data and theoretical biochemistry. Moreover, such models can be applied in a broader framework than that provided by biology. The classical deterministic model of a reaction network is a system of ordinary differential equations, and the standard stochastic model is a continuous-time Markov chain. A relationship between the dynamics of the two models can be found for compact time intervals, while the asymptotic behaviours of the two models may differ greatly. I will give an overview of these problems and show some recent development.

The corner growth model is a classical model of growth in the plane and is connected to other familiar models such as directed last passage percolation and the TASEP through various geometric maps. In the case that the waiting times are i.i.d. with exponential or geometric marginals, the model is well understood: the shape function can be computed exactly, the fluctuations around the shape function are known to be given by the Tracy-Widom GUE distribution, and large deviation principles corresponding to this limit have been derived.

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This talk considers the large deviation properties of a generalization of the classical model in which the rates of the exponential are drawn randomly in an appropriate way. We will discuss some exact computations of rate functions in the quenched and annealed versions of the model, along with some interesting properties of large deviations in this model. (Based on joint work with Elnur Emrah.)

Reconstructing the tree of life from molecular sequences is a fundamental problem in computational

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biology. Modern data sets often contain large numbers of genes. That can complicate the reconstruction because different genes often undergo different evolutionary histories. This is the case in particular in the presence of lateral genetic transfer (LGT), where a gene is inherited from a distant species rather than an immediate ancestor. Such an event produces a gene tree which is distinct from (but related to) the species phylogeny. In this talk I will sketch recent results showing that, under a natural stochastic model of LGT, the species phylogeny can be reconstructed from gene trees despite surprisingly high rates of LGT.

Title: '''A Sample Path Large Deviation Principle for a Class of Population Processes'''

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Abstract: We establish a sample path large deviation principle for sequences of Markov chains arising in game theory and other applications. As the state spaces of these Markov chains are discrete grids in the simplex, our analysis must account for the fact that the processes run on a set with a boundary. A key step in the analysis establishes joint continuity properties of the state-dependent Cramer transform L(·,·), the running cost appearing in the large deviation principle rate function.

Abstract: A conjecture of Fyodorov, Hiary & Keating states that the maxima of the characteristic polynomial of random unitary matrices behave like the maxima of a specific class of Gaussian fields, the log-correlated Gaussian fields. We will outline the proof of the conjecture for the leading order of the maximum, and a freezing of the free energy related to the matrix model. This talk is based on a joint work with Louis-Pierre Arguin and David Belius.

Abstract: We study two rumor processes on the positive integers, the dynamics of which are related to an SI epidemic model with long range transmission. Start with one spreader at site <math>0</math> and ignorants situated at some other sites of <math>\mathbb{N}</math>. The spreaders transmit the information within a random distance on their right. Depending on the initial distribution of the ignorants, we obtain probability of survival, information on the distribution of the range of the rumor and limit theorems for the proportion of spreaders. The key step of our approach is to relate this model to the house-of-cards.

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Abstract: Stochastic homogenization studies the effective equations or laws that characterize the large scale phenomena for systems with complicated random dynamics at microscopic levels. In this talk, we explore the relation between stochastic homogenization and a probabilistic model called random motion in a random medium. In particular we focus on dynamics on the integer lattice which is non-reversible in time and defined by a non-divergence form operator which is non-elliptic. A difficulty in studying this problem is that coefficients of the operator are allowed to be zero. Using random walks in random media, we present a Harnack inequality and a quantitative result for homogenization for this random operator. Joint work with N.Berger (TU-Munich), M.Cohen (Jerusalem) and J.-D. Deuschel (TU-Berlin).

Abstract: Dyson's Brownian Motion may be thought of as a generalization of Brownian Motion to the matrix setting. We can study the eigenvalues of a Dyson's Brownian motion at multiple times. The resulting object has different "color" points corresponding to the eigenvalues at different times. Similar to a single time, the correlation functions of the process may be described in terms of determinantal formulas. We study the local behavior of the eigenvalues as we take the dimension of the associated matrix to infinity. The resulting limiting process in the bulk is again determinantal and is described with an "extended sine kernel." This work aims to give an alternate description of the limiting process in terms of the counting function. In this seminar I will go over the the description and methods for finding such a limit. This is work in progress and is joint with Elliot Paquette (Weizmann Institute).

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[[Past Seminars]]

[[Past Seminars]]

Revision as of 14:41, 30 November 2017

Fall 2017

Thursdays in 901 Van Vleck Hall at 2:25 PM, unless otherwise noted.
We usually end for questions at 3:15 PM.

If you would like to sign up for the email list to receive seminar announcements then please send an email to join-probsem@lists.wisc.edu.

The hard edge refers to the distribution of the smallest singular value for certain ensembles of random matrices, or, and what is the same, that of the minimal point of a logarithmic gas constrained to the positive half line. For any "inverse temperature" and “quadratic" potential the possible limit laws (as the dimension, or number of particles, tends to infinity) was characterized by Jose Ramirez and myself in terms of the spectrum of a (random) diffusion generator. Here we show this picture persists for more general convex polynomial potentials. Joint work with Patrick Waters.

Abstract:
We investigate a simple model for social learning with two characters: a teacher and a student. The teacher's goal is to teach the student the state of the world , however, the teacher herself is not certain about and needs to simultaneously learn it and teach it. We examine several natural strategies the teacher may employ to make the student learn as fast as possible. Our primary technical contribution is analyzing the exact learning rates for these strategies by studying the large deviation properties of the sign of a transient random walk on .

Abstract:
The KPZ fixed point is the Markov process at the centre of the KPZ universality class. In the talk we describe the exact solution of the totally asymmetric simple exclusion process, which is one of the models in the KPZ universality class, and provide a description of the KPZ fixed point in the 1:2:3 scaling limit. This is a joint work with Jeremy Quastel and Daniel Remenik.

Thursday, November 9, 2017, Chen Jia, University of Texas at Dallas

Mathematical foundation of nonequilibrium fluctuation-dissipation theorems and a biological application

The fluctuation-dissipation theorem (FDT) for equilibrium states is one of the classical results in equilibrium statistical physics. In recent years, many efforts have been devoted to generalizing the classical FDT to systems far from equilibrium. This was considered as one of the most significant progress of nonequilibrium statistical physics over the past two decades. In this talk, I will introduce our recent work on the rigorous mathematical foundation of the nonequilibrium FDTs for inhomogeneous diffusion processes and inhomogeneous continuous-time Markov chains. I will also talk about the application of the nonequilibrium FDTs to a practical biological problem called sensory adaptation.

Interacting particle models are often employed to gain understanding of the emergence of macroscopic phenomena from microscopic laws of nature. These individual-based models capture fine details, including randomness and discreteness of individuals, that are not considered in continuum models such as partial differential equations (PDE) and integral-differential equations. The challenge, which is fundamental in any multi-scale modeling approach for complex systems, is how to simultaneously retain key information in microscopic models as well as efficiency and robustness of macroscopic models.

In this talk, I will discuss how this challenge can be overcome by elucidating the probabilistic connections between particle models and PDE, in particular, why naively adding diffusion terms to ordinary differential equations might fail to account for spatial dynamics in population models. These connections also explain how stochastic partial differential equations (SPDE) arise naturally under a suitable choice of level of detail in modeling complex systems. I will also present some novel scaling limits including SPDE on graphs and coupled SPDE. These SPDE not only interpolate between particle models and PDE, but also quantify the source and the order of magnitude of stochasticity. Scaling limit theorems and new duality formulas are obtained for these SPDE, which connect phenomena across scales and offer insights about the genealogies and the time-asymptotic properties of certain population dynamics.

Nonintersecting Brownian bridges on the unit circle form a determinantal point process whose kernel is expressed in terms of a system of discrete orthogonal polynomials which may be studied using Riemann--Hilbert techniques. If the Brownian motions have a drift, then the weight of the orthogonal polynomials becomes complex. I will discuss the tacnode and k-tacnode processes, which are related to the Painleve II function, as scaling limits of Nonintersecting Brownian motions on the unit circle and will discuss some of the features and difficulties of Riemann--Hilbert analysis of discrete orthogonal polynomials with varying complex weights.

Abstract: Stochastic homogenization studies the effective equations or laws that characterize the large scale phenomena for systems with complicated random dynamics at microscopic levels. In this talk, we explore the relation between stochastic homogenization and a probabilistic model called random motion in a random medium. In particular we focus on dynamics on the integer lattice which is non-reversible in time and defined by a non-divergence form operator which is non-elliptic. A difficulty in studying this problem is that coefficients of the operator are allowed to be zero. Using random walks in random media, we present a Harnack inequality and a quantitative result for homogenization for this random operator. Joint work with N.Berger (TU-Munich), M.Cohen (Jerusalem) and J.-D. Deuschel (TU-Berlin).